temperature dependency and temperature compensation in a

Biophysical Chemistry 106(2003) 179–192

0301-4622/03/$ - see front matter� 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0301-4622(03)00191-1

Temperature dependency and temperature compensation in amodel of yeast glycolytic oscillations

Peter Ruoff *, Melinda K. Christensen , Jana Wolf , Reinhart Heinricha, a b b

School of Science and Technology, Stavanger University College, P.O. Box 8002, Ullandhaug, N-4068 Stavanger, Norwaya

Institute of Biology, Humboldt-University Berlin, Invalidenstrasse 42, D-10115 Berlin, Germanyb

Received 17 February 2003; received in revised form 30 June 2003; accepted 1 July 2003

Abstract

Temperature sensitivities and conditions for temperature compensation have been investigated in a model for yeastglycolytic oscillations. The model can quantitatively simulate the experimental observation that the period length ofglycolytic oscillations decreases with increasing temperature. Temperature compensation is studied by using controlcoefficients describing the effect of rate constants on oscillatory frequencies. Temperature compensation of theoscillatory period is observed when the positive contributions to the sum of products between control coefficientsand activation energies balance the corresponding sum of the negative contributions. The calculations suggest that bychanging the activation energies for one or several of the processes, i.e. by mutations, it could be possible to obtaintemperature compensation in the yeast glycolytic oscillator.� 2003 Elsevier B.V. All rights reserved.

Keywords: Temperature compensation; Glycolysis; Biochemical oscillations;Saccharomyces cereviciae; Mathematical modelling;Metabolic control theory; Cellular systems

1. Introduction

Many biological oscillators have important reg-ulatory functions of physiological processes and inthe adaptation of organisms to their environments.Circadian rhythms, for example, have clock ortiming functions w1–4x. Because of this, periodsin these biological clocks are compensated againstenvironmental influences such as temperature, pHand nutritional conditions. Among the environmen-tal compensation mechanisms, temperature com-

*Corresponding author. Tel.:q47-51-831887; fax:q47-51-831750.

E-mail address: [email protected](P. Ruoff).

pensation in circadian clocks is one of the bestdocumentedw1,2,5,6x. Temperature compensationmeans that within a physiological temperaturerange the rhythm’s period length remains nearlyconstant at different(but constant) temperatures,despite the fact that most of the underlying indi-vidual (enzyme-catalyzed) processes are quitedependent on temperature. In general, theQ of10

enzyme-catalyzed processes lies in the range of 2w7x, which means that the reaction rate doubles asthe temperature increases by 108C.

Although the existence of temperature compen-sation of biological clocks is well documented,little is known about how these compensation

180 P. Ruoff et al. / Biophysical Chemistry 106 (2003) 179–192

mechanisms operate. Hastings and Sweeneyw8xwere the first to suggest the existence of opposingreactions controlling the period of biological clockssimilar to the opposing processes occurring in atemperature-compensated mechanical pendulum orin modern temperature-compensated electronicclocks w9x. A theory of opposing reactionsw10,11xcan be formulated on the basis of positive andnegative elements that are apparently present inany reaction kinetic oscillator. A brief descriptionof this theory will be given below. Other proposalsconcerning the origin of temperature compensationhave focused on the importance of diffusion-controlled processes or opposing processes withinenzyme catalysis(for a review see Ref.w9x).

While temperature compensation is advanta-geous for the action of biological clocks, there areonly few examples of temperature-compensatedprocesses within pure chemical or biochemicaloscillators. Skrabalw12x was probably the first toreport a temperature-overcompensated process, i.e.a process where the overall reaction rate decreasesas temperature increases. Besides the early workby Skrabal we are aware of only two other reportsw13,14x which experimentally described and mod-elled temperature compensation of a chemicaloscillator.

Biochemical oscillations, such as the oscillatoryglycolysisw15–19x are important experimental sys-tems for understanding the dynamics and regula-tion of biological rhythms. However, chemical aswell as biochemical oscillators are generally nottemperature-compensated. Because temperaturecompensation is an important aspect in manybiological clock rhythms, we wondered whether itwould be possible, at least in theory, to constructa temperature-compensated biochemical oscillator.As a target system we chose the glycolytic oscil-lator in yeast cells, because the oscillations areeasily generated, and intensively studied. More-over, a wide spectrum of yeast mutants is available.In this work we investigated how temperaturecompensation can occur in a model of glycolyticoscillations in yeast cells developed by some of usw20x. Our results show that a ‘balance’ in activationenergies of opposing processes appears necessaryto observe temperature compensation. Similarobservations have also been made in a theoretical

temperature-sensitivity study of theDrosophilacircadian oscillator w21,22x and the oscillatoryperoxidase–oxidase reactionw23x.

Our theoretical analysis is based on data on thetemperature dependency of glycolytic oscillationsin suspensions of yeast cells by Betz and Chancew24x and by unpublished work of Hemker, Bakker,Teusink, Richard, Westerhoff and van Dam. Thesedata demonstrate that the frequency of the oscil-lations increase with temperature between 5 and40 8C. A corresponding increase in frequency withincreasing temperature has also been observed inyeast cell extracts by Z. Yuan, K. Tsuji and S.C.Muller (unpublished).¨

2. Condition for temperature compensation inreaction kinetic oscillators

Temperature compensation means that the oscil-lator’s period is nearly constant at different tem-peratures as long as the temperature is heldconstant during the oscillations. Experimentally,temperature compensation is normally observedonly in a certain—for the organism physiologicallyimportant—temperature interval, while outside ofthis interval the period may rapidly change oroscillations may no longer be observable. Also ina recently discovered chemical temperature-com-pensated oscillatorw13,14x temperature compen-sation occurs only in a limited temperature interval.

The type of mathematical model we considerhere consists of a set of coupled differential equa-tions describing how the metabolite concentrationsS change in timei

dSis n v (1)ij j8dt i

In these equations, the parametersn denote stoi-ij

chiometric coefficients of the reaction system andv are the reaction rates of the individual processesj

w25,26x. The reaction rates are functions of therate constants and the metabolite concentrations.A reasonable assumption to make is that thetemperature dependence of the rate constantsk isj

described by the Arrhenius-equation

E yRTjksA e (2)j j

181P. Ruoff et al. / Biophysical Chemistry 106 (2003) 179–192

where A is a pre-exponential factor,E is thej j

activation energy of processj, T is the temperaturein Kelvin and R is the gas constant. TheA ’s andj

E ’s may be considered as temperature-independentj

constants. In the case where rapid-equilibriumconstantsK are included in the rate equations, thel

temperature-dependence ofK can be described inl

an analogous way as by Eq.(2), i.e. by substitutingthe activation energy with the enthalpyDH . In0

l

this case the pre-exponential factor can still betreated as temperature-independent and becomesA sexp(yDS yR) w27x.0

l l

The temperature dependence of the oscillator’speriod P will be a function f of the rate andequilibrium constants, i.e.

Psf k (T), K (T) (3)Ž .j l

To find the condition for temperature compensa-tion in a temperature interval around a certainreference temperatureTsT , we start from dPyref

dT at this point

dP ≠f dk ≠f dKj ls q (4)8 8dT ≠k dT ≠K dTj lj l

Using Eq.(2) for calculating the derivatives dk yjdT and inserting the resulting expression(andanalogous expression for dK ydT) into Eq. (4)l

yields

dln P ≠ln f ≠ln f2 0RT s Eq DH (5)j l8 8dT ≠ln k ≠ln Kj lj l

The terms≠ln f y≠ln k and ≠ln f y≠ln K can bej l

considered as sensitivity coefficients which in thecontext of metabolic regulation are called ‘controlcoefficients’C andC , respectivelyw25,28,29x.j l

Temperature compensation of the oscillationsrequires dlnPydTs0. In general, theC ’s dependj

on the kinetic constantsk , such that the conditionj

for temperature compensation from Eq.(5)becomes only approximately valid within a certaintemperature range. For such a local temperaturerange the condition for temperature compensationcan be formulated as

C E(0 (6)i i8i

where index ‘i’ runs over all rate constants and(rapid) equilibrium constants(if present in thesystem). In reaction kinetic oscillators there areoften distinct positive and negative contributionsto the period, which are reflected by positive ornegativeC valuesw10x. These positive and nega-i

tive contributions appear to be necessary(but notsufficient) elements for physico-chemical oscilla-tions to occur. The simultaneous occurrence ofpositive and negative feedbacks, which appear tobe present in all physico-chemical oscillators hasbeen described by Franckw30x as ‘antagonisticfeedback’.

An alternative way to formulate the conditionof temperature compensation, Eq.(6), is to sepa-rate both negative and positive contributions. Thisleads to an ‘antagonistic balance equation’ withthe ‘opposing’ positive(a) and negative(b)contributions on each side of the equation

C E (y C E (7)a a b b8 8a b

Eq. (7) can be considered as a generalization ofthe proposal by Hastings and Sweeneyw8x whereopposing reactions are already contained withinany oscillator modelw10x and need not necessarilylie outside of the oscillator’s reaction network.

3. The model

The mechanism of the oscillatory glycolysis wasfound to be strongly dependent on the kineticproperties of the key enzyme phosphofructokinase(PFK) w31x. PFK is activated by its substrate,fructose 6-phosphate, by its product, ADP and, insome cases, by its second product, fructose 1,6-bisphosphate. ATP acts at higher concentrations asan inhibitor, whereas AMP activates PFK.Although models which describe glycolytic oscil-lations vary in complexityw32–37x, they are most-ly based on one or several of the regulatoryproperties of PFK. The model we consider here isbased on the cooperative inhibition of PFK byATP and was previously used to study the effect


Fig. 1. Reaction scheme for the glycolytic modelw20x showingthe main reactions of anaerobic glycolysis in a yeast celltogether with the influx and outflux of glucose and pyruva-teyacetaldehyde, respectively.

of intercellular coupling on the synchronization ofindividual oscillating yeast cellsw20x. Here weassume that all cells are synchronized and consider,therefore, the reactions in only one single cell. Thereaction network leading to oscillatory glycolysisin a single cell is shown in Fig. 1 and representthe following: J is the input flux of glucose0

through the cellular membrane;v is the reaction1

velocity of the combined reactions of hexokinase,phosphoglucoisomerase and PFK;v is the velocity2

of the glyceraldehyde-3-phosphate dehydrogenasereaction;v is the velocity of the combined reac-3

tions of phosphoglycerate kinase, phosphoglycer-

ate mutase, enolase and pyruvate kinase;v is the4

velocity of the alcohol dehydrogenase reaction;v5

is the velocity of nonglycolytic ATP-consumption;v is the velocity for forming glycerol from triose6

phosphates, whilev is the degradation of extra-7

cellular pyruvateyacetaldehyde andJ is the netflux of pyruvateyacetaldehyde out of the cell.

The model is based on the following set of rateequations:

dSlsJ yv (8a)0 1dt

dS2s2v yv yv (8b)1 2 6dt

dS3sv yv (8c)2 3dt

dS4sv yv yJ (8d)3 4dt

dN2sv yv yv (8e)2 4 6dt

dA3sy2v q2v yv (8f)1 3 5dt

exdS4 swJyv (8g)7dt

with:

q y1w zB EA3C Fv sk S A 1q (9a)x |1 1 1 3D GKy ~I

v sk S N (9b)2 2 2 1

v sk S A (9c)3 3 3 2

v sk S N (9d)4 4 4 2

v sk A (9e)5 5 3


v sk S N (9f)6 6 2 2

exv skS (9g)7 4

A andA , andN andN denote the concentra-2 3 1 2

tions of ADP and ATP, and NAD and NADH,q

respectively. Since several glycolytic reactions areomitted and other reactions are lumped, the modelvariables denote, in some cases, the concentrationsof pools of intermediates, rather than concentra-tions of individual compounds. This concerns thepool of the triose phosphates, glyceraldehyde 3-phosphate and dihydroxyacetone phosphate(vari-ableS ), and the pool of pyruvate and acetaldehyde2

(variable S ). The concentration of glucose is4

represented by the variableS , and that of 1,3-1

bisphosphoglycerate byS . S denotes the concen-ex3 4

tration of the coupling substance in the externalsolution. Furthermore, the lumping process impliesthat the concentrations of some compounds do notappear as separate model variables. The differentialequations forN andA are omitted, because these1 2

concentrations follow from the conservationconditions

N qN sNsconstant (10a)1 2

A qA sAsconstant (10b)2 3

w denotes the ratio of the total cellular volume tothe extracellular volume.K and q are the inhibi-I

tion constants and the cooperativity coefficient ofthe ATP inhibition, respectively. For the transmem-brane fluxes, we use

J sk G (11a)0 0 ex

and

exJsk S yS (11b)Ž .4 4

In these equations,G denotes the concentrationex

of external glucose which is considered to beconstant,k a rate constant which is directly relatedto the permeability of the membrane. For a moredetailed description of this model the reader isreferred to the work by Wolf and Heinrichw20x.

4. Methods

The model’s differential equations (Eqs. (8–9))were solved numerically by use of theFORTRAN

subroutineLSODE w38x. The calculations were per-formed on a Macintosh computer using Absoft’sFORTRAN 77 compiler for the Macintosh, version4.4 w39x. Calculations involving stochastic fitting(see below) were run on a Unix workstation atStavanger University College.

In the stochastic fitting method the activationenergiesE of the rate constantsk (Eq. (2)) andj j

the enthalpyDH of K were varied randomly0K II

until a certain target functionP (T) wastar

approached closely enough by the numericallycalculated periodP(T). This was achieved bycalculating the root-mean-square deviationx

2w zx |xs P T yP T (12)Ž . Ž .tar i iy ~8y

i

for a variety of temperature pointsT for ani

iterative but random choice of activation energies(which are chosen within a certain activationenergy interval). During the iteration only thoseactivation energies were kept that led to a decreasein x. The FORTRAN function RAN1 w40x was usedas the random number generator. For a givenreference temperatureT (see below) and activa-ref

tion energy E , the pre-exponential factorA ofj j

process ‘j’ is calculated by means of theTref

assigned rate constant valuek by use of thej

Arrhenius equation, Eq.(2), i.e.

A sk exp E yRT (13)Ž .j j j ref

5. Results

5.1. Control coefficients

To investigate the conditions for temperaturecompensation by using Eq.(7), we determined thecontrol coefficients C s≠ln f y≠ln k and C sj j KI

≠ln f y≠ln K . First, a reference temperatureT andI ref

a reference state of limit cycle oscillations waschosen with parameters listed in Table 1(andfigure legends) and using for integration initial


Table 1Parameter values of the reference statea

Parameter Value

J0 2.5 mMymink1 100.0 mMymink2 6.0 mMymink3 16.0 mMymink4 100.0 mMymink5 1.28ymink6 12.0 mMymink 1.8ymink 13.0yminq 4.0KI 0.52 mMN 1.0 mMA 4.0 mMw 0.1

Resulting in a period length of 1.17 min. A description ofa

the parameters can be found in Section 3.

Table 2Initial concentrations for reference state oscillations

Compound Concentration(mM)

Glucose 1.187Glyceraldehyde-3-Pydihydroxyacetone-P 0.1931,3-Bisphosphoglycerate 0.050Pyruvateyacetaldehyde 0.115External pyruvateyacetaldehyde 0.077ADP 1.525ATP 2.475NADq 0.923NADH 0.077

concentrations given in Table 2. Then, the periodsfor varying individually thek ’s (including K ) arei I

calculated by keeping all otherk ’s (i/j) constant.j

Fig. 2 shows the relationships lnP vs. lnk fori

those rate constants for which the control coeffi-cientsC varied most with varyingk . The controli i

coefficients for the reference state are listed inTable 3(calculations have been performed for 1%variations and for 10% variations of the rateconstant and of the inhibition constantK . FromI

the ten adjustable parameters only three contributepositively to the period, i.e. when increasingk ani

increase of the period is observed; see Table 3.Among the positive contributions the input fluxJ has the largest influence. From the parameters0

with a negative contribution to the period, the rate

constantk of nonglycolytic ATP consuming pro-5

cesses exerts the strongest effect.

5.2. Temperature dependency of the oscillationsand conditions for temperature compensation

The condition of Eq.(6) or Eq. (7) impliesthat, theoretically, there should be an infinite num-ber of activation energy combinations leading totemperature compensation. We have tested theapplicability of condition given by Eq.(7) usingthe stochastic fitting method described above tomodel the experimental period–temperature rela-tionship and temperature-compensated conditions.

First, we determined by a stochastic fit, a set ofactivation energies which gives a good descriptionof the temperature dependency of the oscillationperiod to the experimental data of Betz and Chance(w24x; shown as open diamonds in Fig. 3) andHemker et al.(unpublished; shown as open squaresand a thick solid line in Fig. 3). The correspondingactivation energies are given in Fig. 5a and in thefirst column of Table 4. Large solid points in Fig.3 show the result of a stochastic fit of the modelto the data by Hemker et al. While experimentalperiod lengths have been observed as low as 0.2min, the model stops showing oscillations atapproximately 0.8 min. Within the oscillatoryrange of the model, however, there is a goodagreement between experiment and calculations.The considerable dominance of negative contri-butions in the balancing sum, Eq.(6), is reflectedby the fact that the period decreases rapidly withincreasing temperature(see8 C E in last row ofi k ii

Table 4).


Table 3Control coefficientsC for the oscillation period in the refer-i

ence state

Control Value Valuecoefficient ("1% variation) ("10% variation)

CJ0q0.69 q0.68

Ck1y0.21 y0.23

Ck2q0.04 q0.06

Ck3y0.01 y0.01

Ck4q0.26 q0.27

Ck5y1.24 y1.27

Ck6y0.26 y0.28

Ck y0.13 y0.12Ck y0.17 y0.15CKI

y0.86 y0.848 C (without K )i i I y1.02 y1.058 C (with K )i i I y1.88 y1.89

Fig. 2. Determination of the control coefficientsC by plotting the logarithm of the period against the logarithm of the varying ratei

constant. The four cases with the greatest variations inC as a function of lnk (insets) are shown in figures(a)–(d). Determinedi i

C values are shown in Table 3.i

Fig. 3 also shows a set of seven series wherethe oscillator’s ability to show temperature com-pensation was tested. Corresponding NADH-oscil-lations are shown in Fig. 4 where part(a) givesthree examples for the model fitted to the experi-mental data whereas parts(b)–(d) show examplesfor temperature-compensated oscillations of series1, 3 and 6. When calculating the balancing sumsfor these sets(last row in Table 4) it is clearlyseen that these sums are now close to zero whichcorresponds to the condition for temperature com-pensation given in Eq.(6).

With a given set of control coefficients there isclearly a certain variability in the activation ener-gies necessary to satisfy Eq.(6) and thus achievetemperature compensation. Since the balancingsum for the model fitted to the experimental datais negative(Table 4) temperature compensationcan either be achieved by decreasing the activationenergies of steps with negative control coefficientsor by increasing the activation energies of stepswith positive control coefficients. This is in accordwith the three different activation energy profiles

shown in Fig. 5b–d which all result in tempera-ture-compensated oscillators with a period ofapproximately 2.5 min. It is seen that a commonfeature of these profiles is the high activationenergyE of the input flux which has a positiveJ0

control coefficient. In fact, increasingE aloneJ0


Fig. 3. Horizontal lines(1)–(7): Temperature-compensatedoscillations with different periods. Reference parameter values:(1) T s300 K, J s1.0 mMymin, k s25 mMymin, k s2.0ref 0 1 2

mMymin, all other parameter values as in Table 1;E ’s arei

given in Table 4;(2) T s300 K, k s25 mMymin, k s2.0ref 1 2

mMymin, k s2.2ymin, all other parameter values as in Table5

1; E s39.8 kJymol, E s17.1 kJymol, E s9.9 kJymol, E sJ k k k0 1 2 3

29.2 kJymol, E s39.2 kJymol, E s14.3 kJymol, E s23.7k k k4 5 6

kJymol, E s31.3 kJymol, E s23.7 kJymol, E s12.3k kappa KI

kJymol; (3) T s305 K, k s25 mMymin, k s2.0 mMymin,ref 1 2

all other parameter values as in Table 1;E ’s are given in Tablei

4; (4) T s312 K, J s4.1 mMymin, k s25 mMymin, k sref 0 1 2

2.0 mMymin, k s1.1ymin, all other parameter values as in5

Table 1; E s45.8 kJymol, E s15.2 kJymol, E s10.5J k1 k0 2

kJymol, E s31.1 kJymol, E s45.4 kJymol, E s13.3k k k3 4 5

kJymol, E s21.2 kJymol, E s32.6 kJymol, E s20.0k k kappa6

kJymol, E s12.3 kJymol; (5) T s312 K, J s4.3 mMymin,K ref 0I

k s25 mMymin, k s2.0 mMymin, k s0.7ymin, all other1 2 5

parameter values as in Table 1;E s42.0 kJymol, E s15.3J k0 1

kJymol, E s10.7 kJymol, E s28.4 kJymol, E s44.1k k k2 3 4

kJymol, E s13.3 kJymol, E s23.0 kJymol, E s29.6k k k5 6

kJymol, E s20.2 kJymol, E s11.7 kJymol; (6) T s318kappa K refI

K, J s4.5 mMymin, k s20 mMymin, k s1.5 mMymin, k s0 1 2 4

150 mMymin, k s0.7ymin, K s0.45 mM, all other parameter5 I

values as in Table 1;E ’s are given in Table 4;(7) T s312i ref

K, J s4.7 mMymin, k s20 mMymin, k s1.5 mMymin, k s0 1 2 4

150 mMymin, k s0.7ymin, K s0.37 mM, all other parameter5 I

values as in Table 1;E ’s are given in Table 4. Open diamonds:i

experimental data by Betz and Chancew24x. Open squares:experimental data by Hemker et al.(unpublished). Thick solidline: smoothed curve of Hemker et al. data. Large solid dots:fit of model to Hemker et al. data;T s286.5 K, k s25ref 1

mMymin, k s2.0 mMymin, all other parameter values as in2

Table 1;E ’s are given in Table 4.i

makes the period of the oscillations less tempera-ture-sensitive as shown in Fig. 6. Moreover, theactivation energy of nonglycolytic ATP-consump-tion (step 5) which is characterized by a strong

negative control coefficient is decreased for tem-perature-compensated oscillations. The same holdstrue for the enthalpy which enters the temperaturedependence of the inhibition constantK . TheI

values of the other activation energies for stepswith lower absolute values of the control coeffi-cients are more variable.

6. Discussion

6.1. Control coefficients and approximate descrip-tion of period

The results shown in Fig. 3 and Table 4 showclearly that the model of glycolytic oscillationscan be easily temperature-compensated with Eq.(6) as an indicator when temperature compensationis expected to occur. The suitability of Eq.(6) topredict temperature compensation has been shownfor several other chemical and biological oscillatormodelsw10,41,42x. On the basis of the performedsensitivity analysis it may be appropriate toapproximate the oscillator’s periodP, Eq. (3), bythe expression

CiB EkiC FP(h k ,«,k st (14)Ž .1 N 02D Gki,0i

which results from a logarithmic expansion of thefunction P(k ,«, k ) around a reference state1 N

characterized by the rate constantsk and thei,0

reference periodt w10,41x. According to the0

summation theorem for control coefficients ofoscillatory periods the following relation isfulfilled

Csy1 (15)i8i

w25,28,43x. Eq.(15) reflects the fact that the periodis a homogeneous function of the rate constants ofdegreey1. For larger deviations of the rate con-stants from their reference values Eq.(14) is onlyapproximately fulfilled. Some(temperature-sensi-tive) parameters(for exampleK ) may not enterI

the rate equations in a homogenous form as mostof the rate constants normally do. In this case, theperiod will still depend on that parameter(as is


Table 4Testing the balancing equation, Eq.(6)

Parameter E , kJymoli E , kJymoli E , kJymoli E , kJymoli E , kJymoli

(experimental)a (series 1) (series 3) (series 6) (series 7)

J0 16.2 43.9 46.1 41.4 51.0k3 44.9 31.7 48.7 30.4 39.2k4 58.7 44.2 38.0 48.1 42.4k1 13.8 16.0 19.1 14.0 23.3k2 60.7 10.4 11.8 10.8 11.9k5 41.2 13.3 17.9 15.1 19.8k6 31.4 23.7 16.2 24.0 15.7k (kappa) 15.9 22.3 19.0 18.7 19.7k 24.3 30.0 32.2 34.1 31.5K b

I 47.0 12.2 11.4 8.6 10.0

8 C Ei k iiy79.0 y2.3 y6.1 y1.8 y3.6

For calculating the balance sums8 C E (last row), values for control coefficients have been used corresponding to a 10%i k ii

change of parameters(cf. Table 3).Fit to Hemker et al. data, Fig. 3(large solid dots).a

In case ofK , E is interpreted asDH .b 0I i KI

Fig. 4. Amplitudes and form of oscillations.(a) Fit to Hemker et al. data(Fig. 3). With increasing temperature the amplitudedecreases as found experimentally(Chance et al.w45x). (b) and (c) The amplitude for the temperature-compensated oscillationsincreases with increasing temperature, but this effect becomes less important for larger period lengths.


Fig. 5. (a) Activation energy profile for the fit to experimental value as shown in Fig. 3. Note the small contribution of the influxparameterJ to the period’s temperature sensitivity.(b)–(d) Different activation energy profiles for temperature-compensated sets0

with periods close to 2.5 min. In(c) we have the same reference parameter values as in(b), but T s300 K instead of 305 K. Inref

(d) we have the same reference parameter values as in(b), but T s290 K instead of 305 K. Note the large contribution of theref

influx parameterJ to the period’s temperature insensitivity.0

the case forK ) but the summation theorem Eq.I

(15) will not be applicable(see Table 3 where thecontrol coefficients ofK have been consideredI

separately from the control coefficients of the rateconstants).

6.2. Temperature range for temperaturecompensation

Because the control coefficientsC (Table 3)i

must be regarded as local quantities characterizingthe effects of small perturbations, predictions fortemperature compensation by Eq.(6) may not beoptimal over the whole temperature range. Forexample, by increasing onlyE as shown in Fig.J0

6 temperature compensation can be obtained for arelatively small temperature range. However, toimprove temperature compensation over a larger

temperature range a ‘fine-tuning’ of all activationenergies is necessary. This can be either donemanually or by the stochastic fit method asdescribed above. Even then, temperature compen-sation is only observed in a certain and finitetemperature range and generally not over the wholeparameter range where oscillations can beobserved. Although this appears at first as aninsufficiency in a model’s performance, such localtemperature compensation is generally observed inbiological clock oscillators. For example, in thecircadian clock rhythm ofNeurospora crassa,temperature compensation is only observed in thetemperature rangef15–30 8C. Above this rangeNeurospora’s circadian period decreases more rap-idly similar to what is normally observed experi-mentally in chemical(or biochemical) oscillatorsw44x.


Fig. 6. Increase ofE makes the period less temperature-sen-J0

sitive. (1) Thick solid line, same system as thick solid line inFig. 3, with E s16.2 kJymol, 8 C E sSsy78.1 (see alsoJ i i i0

Table 4); (2) E s32 kJymol, Ssy68.3;(3) E s64 kJymol,J J0 0

Ssy46.5; (4) E s100 kJymol, Ssy22.0; (5) E s110J J0 0

kJymol, Ssy15.2; (6) E s115 kJymol, Ssy11.8; (7)J0

E s120 kJymol, Ssy8.4.J0

6.3. Amplitude behavior

The amplitudes of the model’s oscillations fittedto the experimental period–temperature relation-ship (Fig. 3a) have been found to decrease withincreasing temperature(Fig. 4a) in agreement withexperimental resultsw45x. On the other hand, wheninspecting the amplitudes of the temperature-com-pensated oscillations(Fig. 4b–d) one sees theopposite trend, i.e. with increasing temperature theamplitudes of the oscillations increase. This effectbecomes less pronounced as the period of thetemperature-compensated oscillations is forced tolarger values. An increase in amplitude withincreasing temperature has been observed in othertemperature-compensated models as wellw10x. Thebehavior can be explained by phase-space argu-ments as the velocity of the trajectory in phasespace increases with increasing temperature, theoscillator must travel through a larger area inphase space to keep the same period. Thus, theamplitude in temperature-compensated systems isexpected to increase with increasing temperature.In their ‘amplitude model’, Lakin-Thomas et al.w46x have studied the consequences of this behav-ior for the resetting of the circadian clock inNeurospora crassa. It may be noted, however, thatLeloup and Goldbeterw47x found in a model for

the circadian rhythm ofDrosophila a situationwith apparently contrary behavior, i.e. a birhythmicstate where the low amplitude oscillations have alarger period than the high amplitude oscillations.In the case of birhythmicity, two stable limit cyclesof distinct period length and amplitude can occurfor the same set of rate constants and otherparameter values. Although the observations byLeloup and Goldbeter might at first sight beconsidered as a counter-example concerning thegenerality of the amplitude model, it has to betested how the amplitudes for each of the twocoexisting limit cycles behave in a temperature-compensated case. As described in this study andin Ref. w10x, it still could be (and in agreementwith the amplitude model) that in the temperature-compensated case the amplitude for at least oneof the coexisting limit cycles increase with increas-ing temperature.

6.4. Glucose transport kinetics

As shown in Fig. 2a and Table 3 glucose influxJ across the plasma membrane has a high influ-0

ence on the period length in the model. Whenallowing a free float of all activation energies andfitting period lengths to those experimentallyobserved for various temperatures(Fig. 3), anactivation energy forE of approximately 16 kJyJ0

mol is obtained(Table 4), which is a little lowerthan the activation energy that can be calculatedfor free diffusion of glucose in water(see forexample Ref.w48x). Experimentally, the kineticsof two glucose transport systems have been studiedin vesicles ofSaccharomyces cerevisiae, one show-ing Michaelis–Menten type kinetics(facilitateddiffusion transport) while the other shows first-order behavior(entry through a pore). In bothcases, the experimentally determined activationenergies lie in the range of 60 and 30 kJymol,respectivelyw49x. This means, our calculated valuefor E differs from the experimental value by aJ0

factor of 0.25–0.5.Reijenga et al.w50x studied the effect of hexose

transport on the dynamics of glycolytic oscillationsin Saccharomyces cerevisiae cells. They found thattransport of hexoses into the cells have a markedinfluence on the oscillations. By partial blocking


the influx of glucose into the cells using maltosethe authors found an increase in frequency withan increase of the glucose influx. The Reijenga etal. in vivo studies are also in agreement withearlier experiments performed on cell extractsw51x,where it was found that the frequency of theoscillations increased with increasing input rate ofsubstrate glucose or fructose. Although the modelstudied here shows a considerable influence of theglucose influx on the period, the experimentalfindings by Reijenga et al.w50x are opposite to theJ -influence in the model, which predicts a0

decrease in frequency with an increase of glucoseinflux J (Fig. 2a). The reason for this discrepancy0

is not clear, but it could be related to the complexregulation of glucose uptake by different hexosetransportersw52,53x or to the type of PFK regula-tion considered in the studied model. Althoughmodels based on activation of PFKw32x or pyru-vate kinasew54x are in agreement with the exper-imental relationship between frequency andglucose influx, a check of frequency control coef-ficients w29,55,56x showed that even more detailedmodels such as the ‘full-scale model’ of Hynne etal. w33x do not reflect the experimentally observednegative effect of the glucose influx on the oscil-latory period.

6.5. Experimental temperature-compensated yeastglycolytic oscillators?

An interesting question is whether it would bepossible to obtain yeast mutants which have aglycolytic oscillator that is less temperature sensi-tive or even temperature-compensated. As indicat-ed by this study and also by experimentsw50x, theinflux of glucose into the cells seems to play animportant role in the control of glycolytic dynam-ics. Therefore, a possible start in the search for atemperature-compensated-glycolytic oscillator maybe to look at the oscillator’s temperature behaviorin hexose transport mutantsw53x.

However, using glycolytic oscillator models asguidelines is somewhat problematic. Mainlybecause the different models have different sensi-tivities against parameter changes and also focuson different aspects of the oscillationsw29,56x. Anextensive comparison between in vivo behavior of

yeast glycolysis and the in vitro kinetic propertiesof the constituent enzymes showed considerablediscrepancies and for only 50% of the enzymesthe in vitro kinetics did describe the in vivo activitysatisfactorily w57x. Although the dynamics of theglycolytic oscillations in yeast are still not com-pletely understood, model calculations in combi-nation with site directed mutation experiments onthe hexose transporter genesw53x could still beuseful in order to make yeast mutants which havea less temperature-sensitive-glycolytic oscillator.

7. Conclusion

We have shown that temperature compensationis possible in a detailed modelw20x for glycolyticoscillations. Using the same strategy, conditionsfor temperature compensation in other glycolyticoscillator models should be obtainable. By affect-ing the kinetics and activation energies for one orseveral component processes with molecular biol-ogy methods it might even be possible to constructyeast mutants which show temperature-compensat-ed glycolytic oscillations. Such mutants may notonly be of importance for an understanding of themolecular processes in glycolytic oscillations, butalso for our understanding of temperature compen-sation in biological clocks.

Acknowledgments

We thank Hans Westerhoff for allowing us touse the work by Hemker et al. and for bringingup the idea of a temperature-compensated glycol-ytic oscillator. We also thank Stefan Muller for¨helpful correspondence and comments. P.R. thanksJay Dunlap, Department of Genetics, DartmouthMedical School for hospitality and support whilea final version of the manuscript was prepared.

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